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     "source": [
      "## M\u00e9todo de Euler\n",
      "Tomamos o problema de Cauchy:\n",
      "$$\\begin{gather}\n",
      "\\dot{x}=f(x,t) \\\\\n",
      "x(t_0) =x_0\n",
      "\\end{gather}$$\n",
      "e um conjunto de n\u00f3s $\\{t_0,t_1,\\dots,t_n\\} \\subset \\mathbb{R}$ tal que $t_i = t_0+i*h$. Chamaremos o n\u00famero real $h$ de passo do processo num\u00e9rico. O objetivo \u00e9 dar um m\u00e9todo num\u00e9rico que produza uma estimativa da solu\u00e7\u00e3o nos n\u00f3s, isto \u00e9, deveremos produzir uma estimativa $\\{x_0,x_1,\\dots,x_n\\}$ tal que $x_i \\approx x(t_i)$ se $x(t)$ for a solu\u00e7\u00e3o do problema de Cauchy."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "O m\u00e9todo mais simples de fazermos esta estimativa \u00e9 o *m\u00e9todo de Euler* que podemos derivar da s\u00e9rie de Taylor\n",
      "$$ \\begin{gather}\n",
      "x_{i+1} \\approx x(t_{i+1}) = x(t_i + h) = x(t_i) + f(x(t_i), t_i).h + \\mathcal{o}(h^2) \\\\\n",
      "x_{i+1} \\approx x_i + f(x_i,t_i)h\n",
      "\\end{gather}$$\n",
      "O que nos d\u00e1 um m\u00e9todo de avan\u00e7o\n",
      "$$ x_{i+1} = x_i + f(x_i,t_i)h $$"
     ]
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     "input": [
      "def edoeuler(func,t0,x0,tf,h):\n",
      "    ''' M\u00e9todo de Euler para equa\u00e7\u00f5es diferenciais ordin\u00e1rias\n",
      "         func - fun\u00e7\u00e3o do lado direito\n",
      "         t0 - tempo inicial\n",
      "         x0 - estado inicial\n",
      "         tf - tempo final\n",
      "         h  - passo do tempo '''\n",
      "    sol = [x0]\n",
      "    told = t0\n",
      "    tnovo = told + h\n",
      "    while tnovo <= tf :\n",
      "        x = sol[-1] + func(sol[-1],told)*h\n",
      "        sol.append(x)\n",
      "        [told,tnovo]=[tnovo,tnovo+h]\n",
      "    return sol"
     ],
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